Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C4×C5×B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| socle | C22×C5 | C2×#120 |
| prime radical | C2×C5×B(4,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2,3,4)(13,14,15,16,17);(9,12);(5,6)(7,9)(8,10)(11,12);(5,7)(6,9)(8,11)(10,12)320 conjugacy classes — view the full character table.
| maximal subgroup of | C5×AGL(1,𝔽5)×B(4,𝔽2) |
| maximal quotient of | C5×AGL(1,𝔽5)×B(4,𝔽2) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14467);